Problem Set 2 - Sequences of Random Variables | ECE 534 | Assignments Electrical and Electronics Engineering

ECE 434 RANDOM PROCESSES SPRING 2003

PROBLEM SET 2 Due Wednesday, February 19

Sequences of Random Variables

Assigned Reading: Chapter 2 and Sections 8.1-8.3 of the course notes. Additional material on

limits for deterministic sequences can be found in Kenneth Ross, Elementary Analysis: The Theory

of Calculus, Sections 7-10 (pp. 24-48). This book is on reserve in the mathematics library in Altgeld

Hall, and it is required for Math 347: Introduction to Higher Analysis: Real Variables.

Reminders: A one hour quiz on probability will be given 7-8 p.m. Monday, February 10, Room

269 Everitt Laboratroy.

Problems to be handed in:

1. The limit of the product is the product of the limits

Consider two (deterministic) sequences with finite limits: limn→∞ xn=xand limn→∞ yn=y.

(a) Prove that the sequence (yn) is bounded.

(b) Prove that limn→∞ xnyn=xy. (Hint: Note that xnyn−xy = (xn−x)yn+x(yn−y) and use

part (a)).

2. Convergence of random variables on (0,1]

Let Ω = (0,1], let Fbe the Borel σalgebra of subsets of (0,1], and let Pbe the probability measure

on Fsuch that P([a, b]) = b−afor 0 < a ≤b≤1. For the following two sequences of random

variables on (Ω,F, P ), find and sketch the distribution function of Xnfor typical n, and decide in

which sense(s) (if any) each of the two sequences converges.

(a) Xn(ω) = nω − bnωc, where bxcis the largest integer less than or equal to x.

(b) Xn(ω) = n2ωif 0 <ω<1/n, and Xn(ω) = 0 otherwise.

3. Convergence of a sequence of discrete random variables

Let Xn=X+(1/n) where P[X=i] = 1/6 for i= 1,2,3,4,5 or 6, and let Fndenote the distribution

function of Xn.

(a)For what values of xdoes Fn(x) converge to F(x) as ntends to infinity?

(b) At what values of xis FX(x) continuous?

4. Sums of i.i.d. random variables, I

A gambler repeatedly plays the following game: She bets one dollar and then there are three

possible outcomes: she wins two dollars back with probability 0.4, she gets just the one dollar back

with probability 0.1, and otherwise she gets nothing back. Roughly what is the probability that

she is ahead after playing the game one hundred times?

5. Sums of i.i.d. random variables, II

Let X1, X2,... be independent random variable with P[Xi= 1] = P[Xi=−1] = 0.5.

(a) Compute the characteristic function of the following random variables: X1,Sn=X1+···+Xn,

and Vn=Sn/√n.

(b) Find the pointwise limits of the characteristic functions of Snand Vnas n→ ∞.

Partial preview of the text

Download Problem Set 2 - Sequences of Random Variables | ECE 534 and more Assignments Electrical and Electronics Engineering in PDF only on Docsity!

ECE 434 RANDOM PROCESSES SPRING 2003

PROBLEM SET 2 Due Wednesday, February 19

Sequences of Random Variables

Assigned Reading: Chapter 2 and Sections 8.1-8.3 of the course notes. Additional material on limits for deterministic sequences can be found in Kenneth Ross, Elementary Analysis: The Theory of Calculus, Sections 7-10 (pp. 24-48). This book is on reserve in the mathematics library in Altgeld Hall, and it is required for Math 347: Introduction to Higher Analysis: Real Variables.

Reminders: A one hour quiz on probability will be given 7-8 p.m. Monday, February 10, Room 269 Everitt Laboratroy.

Problems to be handed in:

The limit of the product is the product of the limits Consider two (deterministic) sequences with finite limits: limn→∞ xn = x and limn→∞ yn = y. (a) Prove that the sequence (yn) is bounded. (b) Prove that limn→∞ xnyn = xy. (Hint: Note that xnyn − xy = (xn − x)yn + x(yn − y) and use part (a)).
Convergence of random variables on (0,1] Let Ω = (0, 1], let F be the Borel σ algebra of subsets of (0, 1], and let P be the probability measure on F such that P ([a, b]) = b − a for 0 < a ≤ b ≤ 1. For the following two sequences of random variables on (Ω, F, P ), find and sketch the distribution function of Xn for typical n, and decide in which sense(s) (if any) each of the two sequences converges. (a) Xn(ω) = nω − bnωc, where bxc is the largest integer less than or equal to x. (b) Xn(ω) = n^2 ω if 0 < ω < 1 /n, and Xn(ω) = 0 otherwise.
Convergence of a sequence of discrete random variables Let Xn = X +(1/n) where P [X = i] = 1/6 for i = 1, 2 , 3 , 4 , 5 or 6, and let Fn denote the distribution function of Xn. (a)For what values of x does Fn(x) converge to F (x) as n tends to infinity? (b) At what values of x is FX (x) continuous? (c) Does the sequence (Xn) converge in distribution to X?
Sums of i.i.d. random variables, I A gambler repeatedly plays the following game: She bets one dollar and then there are three possible outcomes: she wins two dollars back with probability 0.4, she gets just the one dollar back with probability 0.1, and otherwise she gets nothing back. Roughly what is the probability that she is ahead after playing the game one hundred times?
Sums of i.i.d. random variables, II Let X 1 , X 2 ,... be independent random variable with P [Xi = 1] = P [Xi = −1] = 0.5. (a) Compute the characteristic function of the following random variables: X 1 , Sn = X 1 + · · · + Xn, and Vn = Sn/

n. (b) Find the pointwise limits of the characteristic functions of Sn and Vn as n → ∞. (c) In what sense(s), if any, do the sequences (Sn) and (Vn) converge?

Sums of i.i.d. random variables, III Fix λ > 0. For each integer n > λ, let X 1 ,n, X 2 ,n,... , Xn,n be independent random variables such that P [Xi,n = 1] = λ/n and P [Xi,n = 0] = 1 − (λ/n). Let Yn = X 1 ,n + X 2 ,n + · · · + Xn,n. (a) Compute the characteristic function of Yn for each n. (b) Find the pointwise limit of the characteristic functions as n → ∞ tends. The limit is the characteristic function of what probability distribution? (c) In what sense(s), if any, does the sequence (Yn) converge?
Mean square convergence of a random series The sum of infinitely many random variables, X 1 + X 2 + · · · is defined as the limit as n tends to infinity of the partial sums X 1 + X 2 + · · · + Xn. The limit can be taken in the usual senses (in probability, in distribution, etc.). Suppose that the Xi are mutually independent with mean zero. Show that X 1 + X 2 + · · · exists in the mean square sense if and only if the sum of the variances, Var(X 1 ) + Var(X 2 ) + · · ·, is finite. (Hint: Apply the Cauchy criteria for mean square convergence.)
Portfolio allocation Suppose that you are given one unit of money (for example, a million dollars). Each day you bet a fraction α of it on a coin toss. If you win, you get double your money back, whereas if you lose, you get half of your money back. Let Wn denote the wealth you have accumulated (or have left) after n days. Identify in what sense(s) the limit limn→∞ Wn exists, and when it does identify the value of the limit (a) for α = 0 (pure banking), (b) for α = 1 (pure betting), (c) for general α. (d) What value of α maximizes the expected wealth, E[Wn]? Would you recommend using that value of α? (e) What value of α maximizes the long term growth rate of Wn (Hint: Consider log(Wn) and apply the LLN.)
A large deviation Let X 1 , X 2 , ... be independent, N(0,1) random variables. Find the constant b such that

P [X^21 + X^22 +... + X^2 n ≥ 2 n] = exp(−n(b + n))

where n → 0 as n → ∞. What is the numerical value of the approximation exp(−nb) if n = 100.

Problem Set 2 - Sequences of Random Variables | ECE 534, Assignments of Electrical and Electronics Engineering

Related documents

Partial preview of the text

Download Problem Set 2 - Sequences of Random Variables | ECE 534 and more Assignments Electrical and Electronics Engineering in PDF only on Docsity!

ECE 434 RANDOM PROCESSES SPRING 2003